Notes
two semi-circles and an equilateral triangle solution

Solution to the Two Semi-Circles and an Equilateral Triangle Puzzle

Solution by Pythagoras' Theorem and Lengths in Equilateral Triangles

In the above diagram, point $O$ is the centre of the semi-circles. As the diameters of the semi-circles cut the triangle halfway along the two sides, $O$ is the midpoint of $C F$. As the triangle is equilateral, $C A$ has length $4$ so $C D$ has length $2$ and then $O C$ has length $\sqrt{3}$. So the area of the orange semi-circle is $\frac{3}{2} \pi$.

Then $O F A$ is a right-angled triangle with $O F$ of length $\sqrt{3}$ and $A F$ of length $2$, so by Pythagoras' theorem, $O A$ has length $\sqrt{7}$. The yellow triangle therefore has area $\frac{7}{2} \pi$.

The total are of the two semi-circles is then $5 \pi$.